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We present two exact results for singular features in the radio frequency intensity $I(\omega)$ for ultracold Fermi gases.
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We will use the zeroth moment sum rule N σ N_{\sigma} to fix our normalization of I σ ( ω ) I_{\sigma}(\omega) , since we have set the RF coupling constant in (1) to unity
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The quasiparticle weight and effective mass are related to the real self-energy Σ ′ \Sigma^{\prime} via the standard results: Z = 1 / [ 1 − ∂ Σ ′ / ∂ ω ] Z=1/\left[1-\partial\Sigma^{\prime}/\partial\omega\right] and m / m ∗ = Z [ 1 + ∂ Σ ′ / ∂ ϵ k ] m/m^{*}=Z\left[1+\partial\Sigma^{\prime}/\partial\epsilon_{k}\right] evaluated at the Fermi surface: ( k = k F , ω = 0 ) (k=k_{F},\omega=0)
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If m ∗ ≃ m m^{*}\simeq m , but Z < 1 Z<1 , the singular contribution to I ( ω ) I(\omega) is a delta-function at ω ∗ \omega^{*} , and together with a incoherent piece which has the universal high frequency tail
Cited in the paper.
The i q ℓ iq_{\ell} Matsubara sum can be done analytically using the spectral representation. The 𝐪 {\bf q} -integrals in Σ \Sigma and 𝐤 {\bf k} -integrals in (1) have to be done numerically
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We expect that small a σ , e ≡ a ′ a_{\sigma,e}\equiv a^{\prime} will lead to crossover from ω − 3 / 2 \omega^{-3/2} to an ω − 5 / 2 \omega^{-5/2} tail for ω ≳ ℏ 2 / m a ′ 2 \omega\gtrsim\hbar^{2}/m{a^{\prime}}^{2} . In ref. zwierlein2009 1 / ( k F ↑ a ′ ) 2 ≃ 20 1/(k_{F\uparrow}a^{\prime})^{2}\simeq 20 . Such a crossover is consistent with the exact solution in the Bose limit by C. Chin and P. S. Julienne, Phys. Rev. A 71
Cited in the paper.